Admissibility, Unbiasedness, and Nonnegativity in the Balanced, Random, One-Way Anova Model

Abstract

Problems encountered in estimating variance components in mixed-effects analysis of variance (ANOVA) models have stimulated research in linear estimation. Pukelsheim (1976) noted that quadratics in the observations which are invariant to fixed effects follow a linear model in the variance components, and that results in linear estimation theory can then be applied directly to this derived linear model. However, this model has characteristics which do not permit application of earlier results on admissibility of linear estimators by Cohen (1966) and Rao (1976). Olsen, Seely and Birkes (1976) provided results on admissibility and completeness among unbiased linear estimators in a very general setting which encompasses the peculiar structure of linear models for invariant quadratics. In models with two variance components, they described the essentially complete class of admissible unbiased invariant quadratic estimators of linear combinations of the variance components. These results when combined with existing results on unbiased linear estimation, fairly completed the story on unbiased invariant quadratic estimation of variance components.